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2020 | OriginalPaper | Chapter

On Asymptotically Optimal Solvability of Max m-k-Cycles Cover Problem in a Normed Space

Authors : Edward Kh. Gimadi, Ivan A. Rykov

Published in: Mathematical Optimization Theory and Operations Research

Publisher: Springer International Publishing

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Abstract

We consider the intractable problem of finding m edge-disjoint vertex covers in d-dimensional normed space with maximum total weight, such that each of them has exactly k cycles. We construct a polynomial-time approximation algorithm for solving this problem and derive conditions of its asymptotical optimality.

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Literature
1.
go back to reference Arora, S.: Polynomial-programming: methods and applications. In: Reeves, C.R. (ed.) NATO Advanced Study Institutes Series, Series C: Mathematical and Physical Sciences, vol. 19, pp. 1730–178. Reidel, Dordrecht (1975) Arora, S.: Polynomial-programming: methods and applications. In: Reeves, C.R. (ed.) NATO Advanced Study Institutes Series, Series C: Mathematical and Physical Sciences, vol. 19, pp. 1730–178. Reidel, Dordrecht (1975)
2.
go back to reference Baburin, A.E., Gimadi, E.Kh.: On the asymptotic optimality of an algorithm for solving the maximum m-PSP in a multidimensional Euclidean space. Proc. Steklov Inst. Math. 272(Suppl. 1), 1–13 (2011) Baburin, A.E., Gimadi, E.Kh.: On the asymptotic optimality of an algorithm for solving the maximum m-PSP in a multidimensional Euclidean space. Proc. Steklov Inst. Math. 272(Suppl. 1), 1–13 (2011)
3.
go back to reference Barvinok, A.I., Gimadi, E.Kh., Serdyukov, A.I.: The maximum TSP. In: Punnen, A., Gutin, G. (eds.) The Traveling Salesman Problem and Its Variations, pp. 585–608. Kluwer Acad. Publ., Dordrecht (2002) Barvinok, A.I., Gimadi, E.Kh., Serdyukov, A.I.: The maximum TSP. In: Punnen, A., Gutin, G. (eds.) The Traveling Salesman Problem and Its Variations, pp. 585–608. Kluwer Acad. Publ., Dordrecht (2002)
4.
go back to reference Christofides, N.: Worst-case analysis of a new heuristic for the traveling salesman problem. In: Symposium on New Directions and Recent Results in Algorithms and Complexity, p. 441. Academic Press, New York (1976) Christofides, N.: Worst-case analysis of a new heuristic for the traveling salesman problem. In: Symposium on New Directions and Recent Results in Algorithms and Complexity, p. 441. Academic Press, New York (1976)
5.
go back to reference De Kort, J.B.J.M.: Upper bounds and lower bounds for the symmetric K peripatetic salesman problem. Optimization 23(4), 357–367 (1992)MathSciNetCrossRef De Kort, J.B.J.M.: Upper bounds and lower bounds for the symmetric K peripatetic salesman problem. Optimization 23(4), 357–367 (1992)MathSciNetCrossRef
6.
go back to reference Duhenne, E., Laporte, G., Semet, F.: The undirected m-capacitated peripatetic salesman problem. Eur. J. Oper. Res. 223(3), 637–643 (2012) MathSciNetCrossRef Duhenne, E., Laporte, G., Semet, F.: The undirected m-capacitated peripatetic salesman problem. Eur. J. Oper. Res. 223(3), 637–643 (2012) MathSciNetCrossRef
7.
go back to reference Gimadi, E.Kh.: A new version of the asymptotically optimal algorithm for solving the Euclidean maximum traveling salesman problem. In: Proceedings of the 12th Baykal International Conference 2001, Irkutsk, vol. 1, pp. 117–123 (2001). (in Russian) Gimadi, E.Kh.: A new version of the asymptotically optimal algorithm for solving the Euclidean maximum traveling salesman problem. In: Proceedings of the 12th Baykal International Conference 2001, Irkutsk, vol. 1, pp. 117–123 (2001). (in Russian)
8.
go back to reference Gimadi, E.Kh, Tsidulko, O.Yu.: Asymptotically optimal algorithm for the maximum m-peripatetic salesman problem in a normed space. In: Battiti, R., Brunato, M., Kotsireas, I., Pardalos, P.M. (eds.) LION 12 2018. LNCS, vol. 11353, pp. 402–410. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-05348-2_33 Gimadi, E.Kh, Tsidulko, O.Yu.: Asymptotically optimal algorithm for the maximum m-peripatetic salesman problem in a normed space. In: Battiti, R., Brunato, M., Kotsireas, I., Pardalos, P.M. (eds.) LION 12 2018. LNCS, vol. 11353, pp. 402–410. Springer, Cham (2019). https://​doi.​org/​10.​1007/​978-3-030-05348-2_​33
9.
go back to reference Khachay, M., Neznakhina, E.: Approximation of Euclidean k-size cycle cover problem. Croatian Oper. Res. Rev. (CRORR) 5, 177–188 (2014)MathSciNetCrossRef Khachay, M., Neznakhina, E.: Approximation of Euclidean k-size cycle cover problem. Croatian Oper. Res. Rev. (CRORR) 5, 177–188 (2014)MathSciNetCrossRef
10.
go back to reference Khachay, M., Neznakhina, E.: Polynomial-time approximations scheme for a Euclidean problem on a cycle covering of agraph. Tr. Inst. Matematiki i mehaniki UrORAN 20(4), 297–311 (2014). (in Russian) Khachay, M., Neznakhina, E.: Polynomial-time approximations scheme for a Euclidean problem on a cycle covering of agraph. Tr. Inst. Matematiki i mehaniki UrORAN 20(4), 297–311 (2014). (in Russian)
11.
go back to reference Gimadi, E.Kh.: A new version of the asymptotically optimal algorithm for solving the Euclidean maximum traveling salesman problem. In: Proceedings of the 12th Baykal International Conference, Irkutsk, vol. 1, pp. 117–123 (2001). (in Russian) Gimadi, E.Kh.: A new version of the asymptotically optimal algorithm for solving the Euclidean maximum traveling salesman problem. In: Proceedings of the 12th Baykal International Conference, Irkutsk, vol. 1, pp. 117–123 (2001). (in Russian)
12.
go back to reference Gimadi, E.Kh., Rykov, I.A.: Asymptotically optimal approach to the approximate solution of several problems of covering a graph by nonadjacent cycles. Proc. Steklov Inst. Math. 295, 57–67 (2016) Gimadi, E.Kh., Rykov, I.A.: Asymptotically optimal approach to the approximate solution of several problems of covering a graph by nonadjacent cycles. Proc. Steklov Inst. Math. 295, 57–67 (2016)
14.
go back to reference Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.: The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, Chichester (1985)MATH Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.: The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, Chichester (1985)MATH
16.
go back to reference Papadimitriou, C.: Euclidean TSP is NP-complete. Theor. Comput. Sci. 4, 237–244 (1977)CrossRef Papadimitriou, C.: Euclidean TSP is NP-complete. Theor. Comput. Sci. 4, 237–244 (1977)CrossRef
18.
go back to reference Shenmaier, V.V.: An algorithm for the polyhedral cyclic covering problem with restrictions on the number and length of cycles. Tr. IMM UrORAN 24(3), 272–280 (2018)MathSciNetCrossRef Shenmaier, V.V.: An algorithm for the polyhedral cyclic covering problem with restrictions on the number and length of cycles. Tr. IMM UrORAN 24(3), 272–280 (2018)MathSciNetCrossRef
19.
go back to reference Shenmaier, V.V.: Asymptotically optimal algorithms for geometric Max TSP and Max m-PSP. Discrete Appl. Math. 163(2), 214–219 (2014)MathSciNetCrossRef Shenmaier, V.V.: Asymptotically optimal algorithms for geometric Max TSP and Max m-PSP. Discrete Appl. Math. 163(2), 214–219 (2014)MathSciNetCrossRef
20.
go back to reference Barvinok, A.I., Fekete, S.P., Johnson, D.S., Tamir, A., Woeginger, G., Woodroofe, R.: The geometric maximum traveling salesman problem. J. ACM 50(5), 641–664 (2003)MathSciNetCrossRef Barvinok, A.I., Fekete, S.P., Johnson, D.S., Tamir, A., Woeginger, G., Woodroofe, R.: The geometric maximum traveling salesman problem. J. ACM 50(5), 641–664 (2003)MathSciNetCrossRef
21.
go back to reference Serdyukov, A.I.: An asymptotically optimal algorithm for the maximum traveling salesman problem in Euclidean space. Upravlyaemye sistemy, Novosibirsk 27, 79–87 (1987). (in Russian) Serdyukov, A.I.: An asymptotically optimal algorithm for the maximum traveling salesman problem in Euclidean space. Upravlyaemye sistemy, Novosibirsk 27, 79–87 (1987). (in Russian)
Metadata
Title
On Asymptotically Optimal Solvability of Max m-k-Cycles Cover Problem in a Normed Space
Authors
Edward Kh. Gimadi
Ivan A. Rykov
Copyright Year
2020
DOI
https://doi.org/10.1007/978-3-030-49988-4_6

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